Science:Math Exam Resources/Courses/MATH220/December 2010/Question 09 (b)
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Question 09 (b) 

Prove that if then . 
Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you? 
If you are stuck, check the hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! 
Hint 

Using limit laws, this is very easy. One can also proceed directly by the definition and use part (a). 
Checking a solution serves two purposes: helping you if, after having used the hint, you still are stuck on the problem; or if you have solved the problem and would like to check your work.

Solution 1 

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Please rate my easiness! It's quick and helps everyone guide their studies. Using limit laws, we have that

Solution 2 

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Please rate my easiness! It's quick and helps everyone guide their studies. We could do this manually as well. Let . By definition, there exists an M such that for all we have that Choose where the N comes from part (a). Then for all , we have Thus, by definition, . 